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Report 

Characterisitcs of Noise Received by the 
Software Defined Radio 




Department of Electrical Engineering 
Indian Institute of Technology Madras 

December 2012 



Abstract 



To study the characteristics of the noise received by Software defined radio (SDR) 
and the characteristics of signal transimtted from one SDR to the other SDR using 
Intermediate Frequency (IF) and Radio Frequency (RF). 



Contents 



Abstract i 

Contents ii 

List of Figures iii 

1 Noise and its Characterisitcs 1 

1.1 Introduction 1 

1.2 Colored Noise 1 

1.3 Power Spectral Density 2 

1.4 Calculating Power Spectral Density 2 

1.5 Fast Fourier Transform 3 

1.6 Auto-Correlation 3 

1.7 Calculating Auto-correlation 3 

1.8 Autocorrelation of noise received by SDR 3 

1.9 Autocorrelation of noise filtered using an elliptic filter 5 

2 Signal and its Characteristics 7 

2.1 Introduction 7 

2.2 Sine wave generation 7 

2.3 Autocorrelation of received signal 8 

2.4 Conclusion 8 

References 8 



n 



List of Figures 



1.1 Autocorrelaton of noise received using RF 4 

1.2 Autocorrelation of noise received using IF 5 

1.3 Auctocorrelation of SDR noise samples filterd using an elliptic filter 6 

1.4 Autocorrealtion of random noise generated using Matlab and passed 
through an elliptic filter 6 

2.1 Baseband samples before pulse shaping 7 

2.2 Baseband samples after pulse shaping 8 

2.3 Figure showing Received signal of frequency 120 kHz, its PSD and 
autocorrelation 9 



in 



Chapter 1 

Noise and its Characterisitcs 



1.1 Introduction 

Noise may be defined as any unwanted signal that interferes with the communica- 
tion, measurement or processing of an information-bearing signal. Noise is present 
in various degrees in almost all environments. Noise can cause transmission er- 
rors and may even disrupt a communication process; hence noise processing is an 
important part of modern telecommunication and signal processing systems. De- 
pending on its frequency or time characteristics, a noise process can be classified 
into one of several categories as follows: 

1 Narrowband noise: A noise process with a narrow bandwidth such as a 
50/60 Hz hum from the electricity supply. 

2 White noise: Purely random noise that has a flat power spectrum. White 
noise theoretically contains all frequencies in equal intensity. 

3 Band-limited white noise: A noise with a flat spectrum and a limited band- 
width that usually covers the limited spectrum of the device or the signal 
of interest. 

4 Coloured noise: Non-white noise or any wideband noise whose spectrum 
has a non-flat shape; examples are pink noise, brown noise. 

5 Impulsive noise: Consists of short-duration pulses of random amplitude and 
random duration, and autoregressive noise. 

1.2 Colored Noise 

Although the concept of white noise provides a reasonably realistic and mathe- 
matically convenient and useful approximation to some predominant noise pro- 



lcesses encountered in telecommunication systems, many other noise processes 
are non-white. The term coloured noise refers to any broadband noise with a 
non-white spectrum, study of this characteristic of noise is performed by auto- 
correlation of the noise received by the SDR and by the Power Spectral Density 
(PSD) which is Discrete fourier transform of autocorrelated samples. 

1.3 Power Spectral Density 

Power spectral density function (PSD) shows the strength of the variations(energy) 
as a function of frequency. In other words, it shows at which frequencies variations 
are strong and at which frequencies variations are weak. The unit of PSD is en- 
ergy per frequency(width) and you can obtain energy within a specific frequency 
range by integrating PSD within that frequency range. Computation of PSD 
is done directly by the method called FFT or computing autocorrelation func- 
tion and then transforming it. PSD is a very useful tool if you want to identify 
oscillatory signals in your time series data and want to know their amplitude. 

For example let assume we are operating a factory with many machines 
and some of them have motors inside. We detect unwanted vibrations 
from somewhere. We might be able to get a clue to locate offend- 
ing machines by looking at PSD which would give us frequencies of 
vibrations. 

1.4 Calculating Power Spectral Density 

For communications signals, the energy is effectively infinite (the signals are of 
unlimited duration), so we usually work with Power quantities. So, we find the 
average power by averaging over time. 



1 f T/2 
AveragePower = lim — / (a>r(£)) 2 dt 

T^ooT J_ T / 2 



1 f°° 

lim - / (X T (2irf))df (1.1) 

S x (2*f) 



where S x (u) is the Power Spectral Density (PSD). 



1.5 Fast Fourier Transform 

The Discrete Fourier Transform (DFT) is performed using Fast Fourier transform 
(FFT). In Matlab, vector indices for a N-point vector are numbered from 1 to N. 
Starting at this corresponds to the range of - N-l. The frequency resolution 
may be seen from the fact that there are N points spanning to Fs , where Fs is 
the sampling frequency and so the resolution of each component is Fs /N . The 
first component, number 1, is actually the zero-frequency or "DC" component. 
The true frequency of component 2 is 1 X ^ 
The true frequency of component 2 is 2 x j^ 



The true frequency of component N/2+1 is y x ^ 

1.6 Auto-Correlation 

The term correlation means, in general the similarity between two sets of data. 
Auto-correlation is the cross correlation of a signal with itself. It is used generally 
to know the presence of a periodic signal which has been burried under the noise. 

1.7 Calculating Auto-correlation 

The autocorrelation of a sampled singal is defined as the product of 

N-k 

R xx (k) = j^--r^2x(n)x(n-k) ( L2 ) 

n=0 

where k is the lag or delay. The correlation may be normalized by the mean 
square of the signal R xx (0) giving: 

P= Rx~x~W) (L3) 

Note: N is much larger than k, (N>>k) 

1.8 Autocorrelation of noise received by SDR 

The noise received by the SDR using RF was autocorrelated by using the equa- 
tions (1.4) and (1.5) The noise and the autocorrelation of the noise is shown in 
fig.l 



Received noise using RF 



~~i r 




Power Spectral Density 




Frequency (MHz) 



Figure 1.1: Autocorrelaton of noise received using RF 



The first part of the plot is the samples of noise received by the SDR with 
number of samples on X-axis and Amplitude of the samples on Y-axis. It was 
expected that the noise to be random but here we can see that there is certain 
periodicity of samples. 

The second part of the plot is the autocorrelation performed using (1.4) and 
(1.5) on the samples received in the first part of plot. This autocorrelation plot 
in general was expected to be a sine sort of shape for coloured noise. 

The last part of the plot is the PSD of the plot. This plot gives us the power 
of the signal (noise) with frequency. The two spikes in the plot are because of 
certain periodic nature of noise. 



Received noise using IF 




-1DD -■■ 
-120 1 — 



Power Spectral Density 




Frequency (MHz) 



Figure 1.2: Autocorrelation of noise received using IF 

These plots are just like the plots discussed above except that this noise was 
received using IF. 



1.9 Autocorrelation of noise filtered using an el- 
liptic filter 

The received noise samples by the software defind radio were passed through an 
elliptic filter. The filtered noise samples were sent to the autocorrelation function. 
The figure below shows the autocorrelation of the filtered noise samples. 




Figure 1.3: Autocorrelation of SDR noise samples fllterd using an elliptic filter 

The expected autocorrelation of noise samples is shown in fig 1.4. It was done 
by generating random noise samples and then filtering the samples by an elliptic 
filter in Matlab . The filtered samples were subjected to autocorrelation 






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Figure 1.4: Autocorrealtion of random noise generated using Matlab and passed 
through an elliptic filter 



Chapter 2 

Signal and its Characteristics 



2.1 Introduction 

A baseband signal of frequency F Hz is generated and was upconverted to IF and 
transmitted via channel from one SDR to the other SDR. 

2.2 Sine wave generation 

In this model, we have generated a samples of sine wave for different frequencies 
with different sampling rates and was upsampled by 8 in the Matlab R and pulse 
shaped using RRC filter and given to the SDR which upconverts the signal to IF 
for trasmission of the signal via channel from one SDR to the other SDR. The 
plots of the generated sine wave are shown in fig 2.1 



I channel before pulse shaping 




100 150 

Sample Index 



250 



Figure 2.1: Baseband samples before pulse shaping 



The figure above shows the upsampled version of the baseband signal. We 



Pulse shaped I channel 




100 150 

Sample Index 



250 



Figure 2.2: Baseband samples after pulse shaping 

can see that there are zeros between any two samples this is because it is padded 
with zeros for upsampling. 

Fig 2.2 above shows the pulse shaped version of the previous figure. Pulse shaping 
was done using Raise Cosine filter with roll of factor 0.9. 

2.3 Autocorrelation of received signal 

The transmitted samples from the SDR was received by the other SDR. The 
received samples where analyzed using the autocorrelation function as given in 
equation (1.1) and by the Power Spectral Density (PSD) which is the DFT of 
the samples received by the SDR. The DFT is performed using Fast Fourier 
Transform (FFT) in Matlab R . The received signal, its Power Spectral Density 
and its autocorrelation are shown in fig 2.3 

2.4 Conclusion 



From the graphs above we can see that the received samples were almost with 
zero noise and recovered the samples transmitted by the SDR 




Figure 2.3: Figure showing Received signal of frequency 120 kHz, its PSD and 
autocorrelation